mt185-sample1

# mt185-sample1 - z 7→(1 i z 1(b Describe the image of S...

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Math 185 Sample First Midterm [This may be a bit longer than our exam will be.] 1. (20 points) Carefully deﬁne the following. (In each deﬁnition you may use without deﬁning them any terms or symbols that were used in the text prior to that deﬁnition.) (a). Analytic function (b). Closed set (c). ± -neighborhood (d). Branch point (e). z c ( z,c C , z 6 = 0) 2. (10 points) Prove that if f 0 ( z ) = 0 everywhere in a domain D , then f ( z ) must be constant throughout D . In proving this, you should be careful not to use any results from later in the book that rely on this result. 3. (10 points) Find all complex solutions to the equation z 6 - 1 2 + i 2 = 0. You may use polar coordinates. 4. (15 points) Let S = { z C : 0 Re z 1, 0 Im z 2 π } . (a). Describe the image of S under the map
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Unformatted text preview: z 7→ (1 + i ) z + 1. (b). Describe the image of S under the map z 7→ e z . (c). Describe the image of S under the map z 7→ Arg( z + i ). (Arg denotes the principal branch of the argument.) (d). Describe the images of S under the maps z 7→ Arg ( z + i ), where Arg is any branch of the argument deﬁned on S . 5. (15 points) Let f ( z ) = x 3 + i (1-y ) 3 (where z = x + iy as usual). Show that the only point where f is diﬀerentiable is z = i . 6. (15 points) Let f ( z ) = f ( x + iy ) = u ( x,y )+ iv ( x,y ) is entire and that the ﬁrst-order partial derivatives of u and v are continuous. Let h ( z ) = f ( z ). Show that h ( z ) is entire....
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## This note was uploaded on 09/01/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at Berkeley.

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